Startegies and tactics in measure games Grzegorz Plebanek, Piotr - - PDF document

Startegies and tactics in measure games

Grzegorz Plebanek, Piotr Borodulin-Nadzieja Lecce, December 2005

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Game For a family of sets J we consider game BM(J ) with two players (Empty, who starts the game with an element of J and Non- empty). Players have to choose sets from J included in the last move of the adversary. Empty wins if the intersection of the game is empty. Strategy A function σ: J <ω − → J is called winning strategy for Nonempty in BM(J ) if

• n∈ω

Kn = ∅, whenever (Kn)n∈ω is a sequence in J such that Kn+1 ⊆ σ(K0, ..., Kn) for every n.

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Topology If (X, τ) is a topological space, we can con- sider a game BM(τ \ {∅}). It is usually called a Choquet game. It is convenient to say that a topological space X is Choquet if Nonempty has a winning strategy in BM(X). Theorem (Oxtoby) A nonempty topological space X is a Baire space iff Empty has no win- ning strategy in the game BM(X) Corollary Every Choquet space is Baire.

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Measures For a measure µ on Σ we can consider a game BM(Σ+). We will say that a measure µ|Σ is weakly α-favourable if Nonempty has a winning strategy in BM(Σ+). Theorem (Fremlin) Every weakly α-favourable measure is perfect. Explanation A measure (X, Σ, µ) is perfect if for every measurable function f: X → [0, 1] and every E ∈ Borel([0, 1]) such that f−1(E) ∈ Σ we can find a borel set B ⊆ E such that µf−1(E) = µf−1(B).

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Strategy A function σ: J <ω − → J is called winning strategy for Nonempty in BM(J ) if

• n∈ω

Kn = ∅, whenever (Kn)n∈ω is a sequence in J such that Kn+1 ⊆ σ(K0, ..., Kn) for every n. Tactic A function τ: J − → J is called winning tactic for Nonempty in BM(J ) if

• n∈ω

Kn = ∅, whenever (Kn)n∈ω is a sequence in J such that Kn+1 ⊆ τ(Kn) for every n.

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Theorem (Debs) There is a class J for which Nonempty has a winning strategy in BM(J ) but doesn’t have any winning tactic. Example Let Baire be the algebra of subsets

• f [0, 1] with the Baire property, and M the

ideal of meager subsets of [0, 1]. Denote by J the family Baire \ M. Nonempty has a winning strategy in BM(J ). Nonempty doesn’t have a winning tactic in BM(J ).

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Fact Nonempty doesn’t have a winning tactic in BM(J ). Let U be a countable base for the topology of [0, 1], not containing the empty set. Assume for the contradiction that there is a winning tactic for Nonempty in BM(J ). Denote it by τ. For every U ∈ U there is a V ∈ U (good for U) such that for every M ∈ M ∀M ∈ M ∃N ∈ M N ⊇ M V ⊆∗ τ(U \ N). Construct a sequence (Vn)n∈ω such that Vn+1 is good for Vn for every n and

n∈ω Vn contains

at most one point. The sequence (Vn)n∈ω is a framework of the play for which Nonempty’s tactic fails.

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We will say that a measure µ|Σ is weakly α- favourable if Nonempty has a winning strategy in BM(Σ+). We will say that a measure µ|Σ is α-favourable if Nonempty has a winning tactic in BM(Σ+). Problem Is every weakly α-favourable measure α-favourable?

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countably compact = ⇒ α-favourable = ⇒ weakly α-favourable = ⇒ perfect Definition Family of sets K is countably com- pact if for every sequence (Kn)n∈ω of sets from K such that its every finite intersection is non- empty, we have

• n∈ω

Kn = ∅. Definition (Marczewski) Measure µ|Σ is count- ably compact if it is inner regular with respect to some countably compact class K, it means for every E ∈ Σ we have µ(E) = sup{µ(K): K ∈ K, K ⊆ E}.

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Question Does every measure defined on sub- σ-algebra Σ of Borel([0, 1]) need to be count- ably compact? Theorem (Fremlin) Every measure defined on Σ ⊆ Borel([0, 1]) is weakly α-favourable. Theorem (Plebanek, PBN) A measure µ|Σ (where Σ as above) is countably compact pro- vided there is a family {Bα}α<ω1 of analytic sets, such that µ is regular with respect to the family of those E ∈ Σ for which there is α < ω1 such that E ⊆ Bα is closed in Bα. Theorem (Plebanek, PBN) Every measure de- fined on sub-σ-algebra of Borel([0, 1]) is an im- age of a monocompact measure.

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Theorem (Fremlin) Every measure defined on Σ ⊆ Borel([0, 1]) is weakly α-favourable.

• for n ∈ ω and ψ ∈ ωn denote

V (ψ) = {x ∈ N : x(k) ≤ ψ(k) for all k < n};

• let An be n-th move of Nonempty and Bn
• n-th move of his adversary;
• let Fn ∈ Closed([0, 1]×ωω) such that π(Fn) =

An;

• Nonempty will construct inductively collec-

tion of functions (φn)n∈ω from ωω such that µ∗(Yn) > 0, where Yn =

n

• k=0

π(Fk ∩ ([0, 1] × V (φk|n))

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• Nonempty will construct inductively collec-

tion of functions (φn)n∈ω from ωω such that µ∗(Yn) > 0, where Yn =

n

• k=0

π(Fk ∩ ([0, 1] × V (φk|n)) and play the measurable hull of Yn as Bn;

• consider any sequence xn ∈ Yn, which is

convergent (to some x ∈ [0, 1]);

• fix k ∈ ω;
• for every n ≥ k we can find

yn ∈ Fk ∩ ([0, 1] × V (φk|n)) moreover we can assume that (yn)n con- verges to some y ∈ ωω;

• then (x, y) ∈ Fk and thus x ∈ Ak but k was

arbitrary.

References:

• D. H. Fremlin, Weakly α-favourable mea-

sure spaces, Fundamenta Mathematicae 165 (2000).

• G. Debs, Strat´

egies gagnantes dans cer- tains jeux topologiques, Fundamenta Math- ematicae 126 (1985).

• G. Plebanek, PBN, On compactness of mea-

sures on Polish spaces, Illinois Journal of Mathematics 42 (2005), no 2.

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